"The Evans Equations of Unified Field Theory" Rev.3.2

Last update: February 21, 2006

Supplements:

Febr 13: p.138/139 ^{ }

Febr 15: pp.144-175

Febr 16: p.156

Febr 21: p.157

To be supplemented occasionally

Quotes from Felker's book in **black**, comments in **blue**

The book is written for laymen's introduction to Myron W. Evans' CGUFT, nowadays also called
ECE Theory (= EVANS' Cartan Einstein Theory). So certain simplifications and reductions are necessary,
however, the text should be as correct as possible.
Instead it contains a collection of erroneous and strange formulations up to *severe
irremovable* errors.

Who is responsible for that disaster? Of course, first of all the author himself. A lot of akward and cryptic formulations could have been avoided, if the author would have been equal to his task. However, even more responsibility is to Myron W. Evans whose ideas are only reflected by Lar Felker. And Evans expressly recommends Felker's book on his website.

What I have listed below are only the tips of icebergs without any intention of completeness.

The most akward formulations could be removed without consequences,
because the book is not of mathematical stringency. E.g., the remarks on p.26 about
**C = A** × **B**
are simply wrong and could be skipped. The bad formulations, however, give evidence of the
author's mathematical ignorance and are embarrasing therefore.

When reviewing the book one should imagine that poor willing readers try to understand and interprete these enigmatic and misleading formulations. They waste their time and their confusion grows from page to page.

Download of L. Felker's original pdf-text from MW Evans' website (2.8 MByte)

__p.14__

^{4} As this book was being written, it was hard to keep up with developments.
For example, explanation of the *Faraday homopolar
generator which has been a mystery for 130 years* has been published.

The author should have a look into the relevant physics
text books. Then he will detect that the mysteries of the Faraday homopolar generator
are *his own* mysteries only.

__p.26__

**a** × **b** is called the cross product. If **a** and **b**
are *2-dimensional* vectors, the cross product gives a resulting vector in a third dimension

"Perpendicular to both 3-D vectors **a** and **b**" would be correct,
i.e. both **a** and **b** and the resulting vector **a**×**b** are 3-D vectors.

Generalizing to 4 dimensions, if **A** and **B** are vectors in
the xyz-volume, then *vector* **C = A** × **B**
will be perpendicular or “orthogonal” to the xyz-volume.

**WRONG: **A ** vector C** =

The wedge product is the four dimensional cross product.

**The wedge product AÙ B in 4-D space
is no vector.**

__p.35__

Imagine that we take a penny and a truck, we take them out into a region with very low gravity, and we attach a rocket to each with 1 kg of propellant. We light the propellant. Which – penny or truck – will be going faster when the rocket stops firing? We are applying the same force to each. The penny. At the end of say 1 minute when the rockets go out, the penny will have traveled farther and be going faster. It has less mass and could be accelerated more. If instead we drop each of them from a height of 100 meters, ignoring effects of air resistance, they will both hit the ground at the same time. Gravity applies the same force to each, as did the rocket. Thus, gravity seems like a force but is not a force like the rocket force. It is a false force. It is actually due to curved spacetime. Any object – penny or truck – will follow the curved spacetime at the same rate. Each will accelerate at the same rate. If gravity were a force, different masses would accelerate at different rates.

**An erroneous example**: The author should apply his
"theory" to 1 penny compared with a bag of 1.000.000 pennies (instead
of the truck).

__p.47__

The tetrad is introduced here and will be developed further in later chapters.
The tetrad *is a 4×4 matrix* of 16 components that are built
from vectors in the base manifold and the index as shown in Figure 2-9.

NO! A tetrad is a "4-bein" (tetra = 4), i.e. a ** set of 4
vectors e_{0},e_{1},e_{2},e_{3}
in the tangent space** of each point of the spacetime manifold
with certain additional geometric properties. See the remarks to p.49.

__p.49__

It is built from four 4-vectors in the base manifold - our real universe - and four 4-vectors in the mathematical tangent space.

No, * both* sets of vectors belong to the
tangent space. One set is given by the tangent vectors of the coordinate axes
(usually denoted by

**e _{a}** = q

The coefficients q_{a}^{μ} and q_{μ}** ^{a}**
of these representations can be called "tetrad coefficients",

**V** = V^{μ} **∂**_{μ}
= V^{a} **e _{a}** .

The coordinates of a vector **V**, V^{μ} and V^{a} respectively,
relative to the two reference systems are linearly interrelated with the tetrad coefficients
as coefficients:

V** ^{a}** = V

NB. The tetrad-matrices (q_{μ}** ^{a}**) and (q

__p.51__

Four vectors make up what we call a "4-vector".

No! Each element of a 4-dimensional linear vector space is a "4-vector." That is a result of Felker's permanent misunderstanding of the vector concept. A vector

**V** = V^{μ} **∂**_{μ}
= V^{a} **e _{a}** .

has different four coordinates (not "vectors")
V^{μ} (μ=*0*,*1*,*2*,*3*)
and V^{a} (a=**0**,**1**,**2**,**3**)
relative to different frames.
Felker *erroneously* considers these coordinates as "vectors" itself; hence he can state
«Four *vectors* make up what we call a "4-vector».

Which is the meaning of the η's in (3)?
On p.52 we learn:

A four-vector is multiplied by the *factors* –1, 1, 1, 1 to find the metric four-vector.

**e _{a}**

where η_{0}=−1 and η_{a}=+1 for **a**=1,2,3.
Completely misleading.

What Felker really wants to introduce here is a third kind of vector description: Let
the matrix (η_{ab}) be the Minkowski matrix
diag(−1, 1, 1, 1). Then for arbitrary vectors
**V** = V^{a} **e**_{a} Felker's "metric" four-vector is given by
V_{a}= η_{ab}V^{b}. In other words: Felker's "metric" four-vector
is given by the covariant representation of **V**.

Due to his erroneous misunderstanding of the vector concept Felker does not remark that he is always involved with the same vector.

__p.52__

Completely awful!

__p.61__

ψ(**r**,t) is the wave function of a particle defined over **r** space and
t time. If squared it becomes a probability. In quantum theory ψ²
is used to find the probability of an energy, a position, a time, an angular momentum.

ψ(**r**,t) as solution of the Schroedinger equation
is complex-valued in general. How can it be that ψ² is a probability,
i.e. real-valued?

__p.83__

Figure 4-4

The dot or scalar product is a scalar, just a *distance* without direction.
It can be used to get work or other values also.

The scalar product yields **no distance**
in general. The Figure 4-4 is misleading nonsense.

__p.84__

Two four-vectors define a four dimensional space and the product of two is in the space.

How that? They (**A **and** B**) define
a **two**-dimensional space, the space of all of their linear combinations a**A**
+ b**B**, with the coefficients a,b.

Which product? The scalar or dot product would yield scalars (i.e.no 4-D vectors), the cross product is not defined
and the wedge product yields no vectors, and also the outer product (see p. 89-90) doesn't fit. So what?

If an inner product is defined for every point of a space on a tangent space, all the inner products are called the Riemann metric.

OOOOOH!!! How awful! Do whatever you like, however, don't write a book on a topic you have not understood.

__p.85__

The inner product of two tetrads is: q_{μν} =
q^{a}_{μ} q^{b}_{ν} η_{ab}.
This is a tensor, the symmetric metric, g_{μν},
which is the *distance* between two points or events in 4-dimensional Riemann
space – the space of our universe.

What is the inner product of TWO tetrads. From
where do we obtain two (different?) tetrads? And the symmetric metric should be the
*distance* between two points or events. What a nonsense!

If three vectors are multiplied, the result is a *volume* in three dimensions.
See Figure 4-5.

WRONG! Not for the double cross-product **a**×(**b**×**c**)
as displayed on p.86 Figure 4-6.

__p.86__

Figure 4-6

The scalar or mixed product is a volume.

OK, but that volume is not given by the double
cross-product **a** × (**b** × **c**)
displayed in that figure.

__p.87__

**Divergence**

The divergence of a vector tells the amount of flow *through a point*.

So calculate div (**a**×**r**) where **a**
is constant and **r** denotes the radius vector.

Is there any flow of **F**=**a**×**r
**"through a point" or not?

__p.95__

Evans uses this to decompose the tetrad into parts. Those parts define

gravitation and electromagnetism. He uses a purely mathematical function that

is well known and generalizes to the physical process.

q_{μ}^{a}
= q_{μ}^{a (S)}
+ q_{μ}^{a (A)}

= (symmetric) + (antisymmetric)

= gravitation + electromagnetism

= curvature + torsion

= distance +
turning

**WRONG!!!**
The matrix of the tetrad coefficient doesn't transform symmetrically
relative to the two indices μ and a. Thus the well known splitting of a matrix Q
(cf. p.295: Antisymmetric tensor)

into its symmetric part
Q(S) = ½ [Q + Q^{T}]

and its antisymmetric part
Q(A) = ½ [Q − Q^{T}]

where Q^{T} denotes the transposed matrix of Q,
does not apply here. The reason is that both
matrices Q and Q^{T} have *different*
transformation behavior (co- and contravariantly with respect to the index μ), and hence the addition
of both matrices doesn't yield
a quantity with physical meaning
(apples and pears cannot be added), while the lines

= gravitation + electromagnetism

= curvature + torsion

= distance + turning

contain physical/geometrical quantities. Cf. the chart on p.175.

__p.96__

The Tetrad q^{a}_{μ}

. . . . . . . .

In the tetrad matrix each element is the product
of two vectors. One vector is in the base manifold and one in the
index. q^{0}_{0} indicates
q^{0} times q_{0}.

Which "vectors" could be meant by q^{0} and q_{0} ?
Felker cannot distinguish between vector and vector coordinate (see the remarks to p.51).
Here "vector" means "vector coordinate"
q^{0} or q_{0} respectively.
So we have an outer product of two vectors in the sense of p.89/90 under consideration
yielding the tetrad matrix (q^{a}_{μ}).
However, such a matrix cannot be invertible since having parallel line vectors, in contradiction
to the necessary invertibility of the tetrad matrices (see the remarks to p.49).

__p.97__

Let V^{a} be the four-vector in the tangent orthonormal flat space and V^{μ} be
the corresponding base four-vector, then:

V^{a} = q^{a}_{μ} V^{μ}
(7)

q^{a}_{μ}
is then the tetrad matrix. Here a and μ stand for 16 vectors as shown
in the matrix above.
Each element of q^{a}_{μ}, the tetrad, is V^{a} V^{μ}.

The sentence "Here a and μ stand for 16 vectors as shown in the matrix above" means that the tetrad matrices are 4×4 and therefore have 16 numbers (not vectors) as entries.

May we write the last sentence as the equation
q^{a}_{μ} = V^{a} V^{μ} or what? Then the author should explain
how that could be deduced from Equ.(7); i.e. from its written out version

V^{a}
=
q^{a}_{0} V^{0}
+ q^{a}_{1} V^{1}
+ q^{a}_{2} V^{2}
+ q^{a}_{3} V^{3}

where the indices written in *italics* refer to the coordinate basis of rhe tangent space.
Until we get the author's reply we declare the sentence
"Each element of q^{a}_{μ}, the tetrad, is V^{a} V^{μ}"
to be *nonsense*. And he should explain what is meant by
"a and μ stand for 16 vectors".

Here is only **one** vector, namely **V**,
which has 4 coordinates
V^{0},V^{1},V^{2},V^{3}
relative to the coordinate basis
**∂**_{0},**∂**_{1},**∂**_{2},**∂**_{3}
and other 4 coordinates
V^{0},V^{1},V^{2},V^{3}
relative to the tetrad basis
**e**_{0},**e**_{1},**e**_{2},**e**_{3}:

**V** = V^{μ}**∂**_{μ} = V^{a}**e**_{a} .

The two different vector coordinate sets are interrelated by the equations (7).

Most of the trouble above is caused by the fact that the author confounds a vector with its coordinates relative to different vector bases.

__p.109__

Figure 4-18 Vector Multiplication

contains several enigmatic and misleading formulations: Which is the meaning of the geometrical symbols?

E.g.:
What is meant by

**↑ · ↑ · η**
= distance ?

Perhaps V^{a}W^{b}η_{ab} for two vectors
**↑ = V** = V^{a}**e**_{a},
**↑ = W** = W^{a}**e**_{a}
and the Minkowski matrix **η** = diag(−1, 1, 1, 1) ?

However, that is *no distance* at all; even in the case **V** = **W**
we would obtain the *square* of the Minkowskian length of **V**, namely the value

−
V^{0 2;} + V^{1 2;} + V^{2 2;} + V^{3 2;}

And what shall the reader learn from

Hodge Dual ***A** = dual of **A**

Dual of a 3-vector is a plane.
?

__p.137/138__

The most general asymmetric metric tensor is defined by the outer or tensor product of two tetrads: Here the tetrad is equivalent to metric four-vectors

q^{ab}_{μν}
= q^{a}_{μ} q^{b}_{ν} ,
q^{ab}_{μν}
= q^{ab}_{μν}^{(S)} + q^{ab}_{μν}^{(A)}
(2)

An asymmetric form has both symmetry and antisymmetry within it.

None of the above quantities can be identical with the metric tensor g_{μν}
of spacetime as they have *two additional upper tetrad indices* a,b, which are missing at
g_{μν}.
The *correct* relation is

g_{μν}
= η_{ab} q^{a}_{μ} q^{b}_{ν}.

Here, due to the symmetry of (η_{ab}) = diag(−1 , 1, 1, 1)
the matrix g_{μν} turns out to be symmetric.

The anti-symmetric metric tensor is q^{ab}_{μν}^{(A)}.
This defines an area, dA, with rotational potential.
The antisymmetric metric tensor is defined by the wedge product of two tetrads:

q^{ab}_{μν}^{(A)}
= q^{a}_{μ}Ù q^{b}_{ν}
(3)

Instead of (3) we have to consider

(3')
η_{ab} q^{ab}_{μν}^{(A)}
= η_{ab} q^{a}_{μ}Ù q^{b}_{ν}

which vanishes due to the symmetry of η_{ab}.

Symmetry indicates centralized potentials – spherical shapes.

Anti-symmetry always involves rotational potentials – the helix.

Asymmetry indicates both are contained in the same shape.

__p.144__

The Evans solution,
R^{a}_{μ} – ½ Rq^{a}_{μ}
= kT^{a}_{μ},
is more fundamental than Einstein’s equation,
G_{μν}
= R_{μν} − ½ g_{μν} R = − kT_{μν}.

That is not true. As will readily be shown here both equations are *equivalent*.

**Proof**
We "multiply" the Einstein equation by q_{b}^{ν} η^{ab}
to obtain Evans' equation. Conversely "multiplying" Evans' equation by
η_{ab} q^{b}_{ν} yields the Einstein equation,
i.e. both equations are *equivalent*.
That means:

It is

__p.146__

The electromagnetic and weak fields are described by a postulate similar to Einstein’s R = kT:

A^{a}_{μ} = A^{(0)} q^{a}_{μ}
(12)

A^{(0)} is a fundamental electromagnetic potential in volts-seconds per meter.

cf. Remarks to p.175

__p.151/152__

The Evans Lemma gives quantum field/matter theory from general
relativity. Gravitation is described by the Lemma when the field is the tetrad,
q^{a}_{μ}.

The other three fields – electromagnetism, weak, and strong - are described when the field is the tetrad multiplied by an appropriate scaling factor and is in the appropriate representation space. For example, the fundamental electromagnetic field has the antisymmetry of equation (6) and is described by equation (12).

Equation (6) on p.140 is

g^{ab}_{μν}
^{(A)} = q^{a}_{μ}
_{^} q^{b}_{ν}
(6)

Evans felt remembered of the antisymmetry of the electromagnetic
fundamental tensor F_{μν} = − F_{νμ}

The strong and weak fields are described respectively as:

S^{a}_{μ}
= S(0) q^{a}_{μ}
(13)

W^{a}_{μ} = W(0) q^{a}_{μ}
(14)

Electromagnetism is defined by the torsion form. It is spinning of spacetime itself, not an object imposed upon the spacetime. The weak field is also a torsion form. It is related to electromagnetism. Thus in the neutron's conversion to a proton the torsion form will be involved. We see an electron leave the neutron and a proton remains. Now it is more clear that there was an electrical interaction that has an explanation. See Chapter 12 on the electroweak theory.

That all may be or not: Pure speculations. see p.175

__p.156__

"Multiplication" of Einstein's field equation

G_{μν}
= R_{μν} − ½ R g_{μν} = kT_{μν} .
(EFE)

by q_{b}^{ν} η^{ab}
yields the eqivalent equation

G^{a}_{μ} = R^{a}_{μ} – ½ Rq^{a}_{μ}
= kT^{a}_{μ} .
(9,16)

Multiplication by q^{b}_{ν} leads to Evans' "General Field Equation"

G^{a}_{μ} q^{b}_{ν}
= kT^{a}_{μ} q^{b}_{ν} ,

which due to the invertibility of (q^{b}_{ν}) is eqivalent to the
former versions of Einstein's field equation.
The "multiplication" by η_{ab} leads back to the Einstein equation (EFE).

In addition Evans considers the antisymmetric part of (4.3) to be obtained by wedging
(9,16) by q^{b}_{ν}:

G^{a}_{μ}Ùq^{b}_{ν}
= kT^{a}_{μ}Ùq^{b}_{ν} .
(8)

Now Evans has a **problem**: There are two upper indices a and b (=0,1,2,3) where not any should be.
His way out is contracting
G^{a}_{μ}Ùq^{b}_{ν}
and
T^{a}_{μ}Ùq^{b}_{ν}
to
G^{c}_{μν} and T^{c}_{μν}
respectively.

G^{c}_{μν} = k T^{c}_{μν} .
(p.156)

And specifying *nothing* about c.

**Remark** The only correct way of removing the indices a,b is to "multiply" by η_{ab}.
Due to
η_{ab} q^{a}_{μ}Ùq^{b}_{ν}
= 0
that yields the very convincing equation 0 = 0.

__p.157__

Evans tried at least twice to derive his "Evans wave equation",
in the Chapters 4.2 (p.67 f.) and 9.2 (p.178 f.) of his GCUFT book I.
However, both attempts are wrong due to elementary calculation errors.
Even an unlearned reader can recognize that the equations (4.21-23) on p.69/70 of [2]
cannot be true due to wrong occurrence of the indices λ, μ and ν in
these equation: A formal (and very helpful) condition of tensor calculus is,
that at each summand
an index can at least appear *twice*, *one in the lower and the other one
in the upper index position*. So check the equations (4.21-23) on p.69/70 of [2]!
And a correct calculation yields a result that does not agree with Evans' ideas.

See http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansWaveEqu.html

The result:

__p.174__

Of great interest is that the symmetries of the matrix in mathematics are applied to physical quantities:

q^{a}_{μ}
= q^{a}_{μ}^{(S)} + q^{a}_{μ}^{(A)}

Gravitation can be symmetric or antisymmetric, however it is always curving.

Electromagnetism can be symmetric or antisymmetric, however it is always turning.

The wave equation of general relativity and unified field theory is:

(o + kT) q^{a}_{μ} = 0

By substituting appropriate representations for the tetrad, the various equations of physics can be derived. Gravitation, electromagnetism, the weak force, and the strong force can all be represented. This simple looking equation can be expanded as given in the chart at the end of the chapter. The wave equation was derived from general relativity using differential geometry.

The factorization of the symmetric and antisymmetric metrics from the
asymmetric tetrad is basic differential geometry. It gives four forces – a
discovery opening new insights into physics. This is
q^{a}_{μ}
= q^{a}_{μ}^{(S)} + q^{a}_{μ}^{(A)}.

Applied to physics, four potential fields are represented as shown in the chart.

The standard model is not generally covariant – it does not allow calculations of interactions among particles in different gravitational fields, say near a black hole. The standard model does not allow for electromagnetism's and gravitation's mutual effects to be defined.

__p.175__

The Evans unified equations allow both these processes to be accomplished.

The use of the mathematical representation space for the tangent
spacetime “a” is of the essence. This is the Palatini variation of gravitational
theory. The tetrad can now be expressed as any of the four fields of physics – G,
A, W, and S. Thus four forms of energy can be described within the unified field
q^{a}_{μ} :

While the structure is not fully developed yet, the logic is clear. Gravitation is centralized and symmetric. Mass curves space in a spherical shell around it. The electromagnetic wave, the photon, is antisymmetric spinning spacetime. The electron (charge) is centralized spin. Antisymmetric curvature is a matter for study.

TYPE POTENTIAL FIELD

Gravitational curvature

Symmetric = Centralized
q^{a}_{μ}^{(S)}

Einstein gravitation

Gravitational curvature

Antisymmetric = turning

Unexplored. The strong
q^{a}_{μ}^{(A)}

field, dark matter?

Electromagnetic Field

Antisymmetric EM
A^{a}_{μ}^{(A)}
= A^{(0)} q^{a}_{μ}^{(A)}

Photon, EM waves

Electrodynamics

Symmetric EM
A^{a}_{μ}^{(S)}
= A^{(0)} q^{a}_{μ}^{(S)}

Charge, the electron

MW Evans felt motivated to these four Ansatzes by the shining example of the Einstein
equation G_{μν} = − k T_{μν} (cf. p.146).
However, the ansatzes have a shortcoming that is not present at Einstein's formula: Each potential
should generate a single physical field only.
But there is the tetrad dependent index a (= 0, 1, 2, 3), which gives rise to four fields at once.
More, the generated fields depend on the choice of the tetrad,
i.e. they are *not well determined*. Last not least the physical motivation of the potentials
is questionable.

__p.184__

We have seen that G q^{a}_{μ}
= kT q^{a}_{μ}. This says that curvature, G, and stress
energy, T, are related through the tetrad matrix.
The unified field is the tetrad q^{a}_{μ}.

That's **intellectual fraud**: Due to EINSTEIN we know the equation G = kT.
To multiply that by the factor q^{a}_{μ} is trivial. And saying
that both G and T are related by q^{a}_{μ} is **rubbish**!
G and T were related by Einstein *without Evans' factor* q^{a}_{μ}.

The tetrad is an asymmetric square matrix. This can be broken into its symmetric and antisymmetric parts:

q^{a}_{μ} = q^{a}_{μ}^{(S)}
+ q^{a}_{μ}^{(A)}
(3)

See "Antisymmetric tensor" in Glossary: There a double contravariant tensor is split
into its symmetric and its antisymmetric part. Both parts are double contravariant again.
The same procedure applied to the tetrad matrix (q^{a}_{μ}) would
destroy the type of the matrix, i.e. its transformation behavior, completely,
because of the different characters of upper and lower indices.

__p.296__

Unit vectors **e**_{(1)}, **e**_{(2)},
**e**_{(3)}, **e**_{(4)} which define a mathematical tangent space.

The basis vectors of a reference frame are a group of four mutually orthogonal vectors. Basis vectors establish a unit vector length that can be used to determine the lengths of other vectors. They obey:

**e**_{0}² + **e**_{1}² + **e**_{2}² + **e**_{3}²
= +1

That is, they form a four dimensional sphere. In addition:

**e**_{0}**·****e**_{0} = −1 ,
**e**_{1}**·****e**_{1}
= **e**_{2}**·****e**_{2}
= **e**_{3}**·****e**_{0} = +1

**e**_{a}**·****e**_{b} = 0 if a ≠ b; a, b are 0, 1, 2, or 3.

OK. That could be taken as the definition of the dot product for the linear space spanned by
the vectors
**e**_{0},
**e**_{1},
**e**_{2},
**e**_{3},
i.e. in the tangential space of the spacetime-manifold at some of its points.
The vectors **e**_{1}, **e**_{2}, **e**_{3} are defined on p.297
by means of
the parameter representation **R** = **R**(u_{1},u_{2},u_{3})
(of the spacetime manifold???). By that way we see that the orthogonality relations

**e**_{a}**·****e**_{b} = 0 if a ≠ b; a, b are 0, 1, 2, or 3.

Following that question we shall recognize that
the parameter representation **R** = **R**(u_{1},u_{2},u_{3})
is * not sufficient to represent the spacetime manifold* under consideration.

__p.297__

Orthonormal means they are both orthogonal and normalized. If one finds the orthonormal
vectors at an event point, then one can define the vectors at any other event point in the spacetime.
The spacetime is curved and some method of calculating and visualizing the spacetime is needed.
Unit vectors are tangent to a curve at a point. For example the three unit vectors of a
curved coordinate system are mutually orthogonal ("perpendicular") and cyclically symmetric with
O(3) symmetry. With **e**_{(1,2,3)} the unit basis vectors and u_{(1,2,3)}
the coordinates at any point.

**e**_{(1)}**·****e**_{(2)} = 0,
**e**_{(1)}**·****e**_{(3)} = 0
and
**e**_{(2)}**·****e**_{(3)} = 0

and

**e**_{(1)}×**e**_{(2)} = **e**_{(3)},
**e**_{(1)}×**e**_{(3)} = **e**_{(2)}
and
**e**_{(2)}×**e**_{(3)} = **e**_{(1)}

**e**_{(n)} = 1/h_{i} × ∂**R**/∂u_{i}
read **e**_{i}
= 1/h_{i} × ∂**R**/∂u_{i}

and the arc length is:

ds = |d**R**|
= |∂**R**/∂u_{1}×du_{1}
+ ∂**R**/∂u_{2}×du_{2}
+ ∂**R**/∂u_{3}×du_{3}|

By squaring the last equation one would obtain

ds² = |d**R**|²
= ∂**R**/∂u_{j} du^{j}
**·** ∂**R**/∂u_{k} du^{k}
= g_{jk}du^{j}du^{k}

to be summed w.r.to j,k over 1,2,3. That is merely the metric of a *three*-dimensional
manifold while a metric of the *four*-dimensional spacetime has to be defined.
Not even the case of the (flat) Minkowskian spacetime can be treated by that line element ds.

To which space X does

By the representation

Evans error: Instead he tries to define a dubious pseudo-Euclidean metric; see MWE's difficulties concerning his metric below.

__p.329__

M(A) = M(S) + M(A-S)

Asymmetric equals Symmetric plus Anti-Symmetric

The splitting above from M(A) = M(S) + M(A-S) is no splitting into symmetric and antisymmetric parts.

__p.331__

ds² = g_{μν} x^{μ}
x^{ν} = q^{μ}q^{ν}

read ds² = g_{μν} dx^{μ}dx^{ν}
(= q^{μ}q^{ν} ?).

The latter part in (...) is evidently wrong as it depends on the *summation* indices μ and ν.
Probably the author has meant the equation

ds² = g_{μν} dx^{μ}dx^{ν}
= q_{μ}q_{ν} dx^{μ}dx^{ν} .

However, the last equation would imply that the matrix must have a vanishing determinant, because
all line vectors are parallel to the line [q_{0} q_{1} q_{2} q_{3}].

The metric tensor is defined as the outer product of two metric vectors. Form the outer product of two four vectors, i.e. multiply a column four vector by a row four vector, and you have a 4×4 matrix, with sixteen components.

However, in that case we obtain det(g_{μν}) = 0 due to the above argument.

. . . The symmetric metric is always defined as the tensor or outer product of two vectors.

We have quite generally det (**A**^{T}Ä**B**) = 0
for two arbitrary n-vectors **A**, **B** and their outer product
**A**^{T}Ä**B**. However, the matrix g = det(g_{μν})
is not allowed to vanish since the metric matrix must be invertible. Thus:

__p.332__

We want a real number to define the distance. The metric tensor takes two vectors which define the curvature that occurs and calculates the real distance between them. We cannot have a negative distance so the metric must be "positive definite" – it must be real.

**WRONG!** Even the special metric of Minkowkian spacetime (cf. p.333),
the matrix (η_{μν}), is *indefinite*,
i.e. ds² = − c²dt² + dx² + dy² + dz²
attains *negative* values inside the light cone and *positive* values outside
while it vanishes at the cone itself. And the determinant of the Minkowskian metric,
det(η_{μν}) has the value −1. Thus:

[1]
Laurence G. Felker, *The Evans Equations of Unified Field Theory*; Rev. 3.2 Oct.2005
http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[2]
Myron W. Evans, *GENERALLY COVARIANT UNIFIED FIELD THEORY:
THE GEOMETRIZATION OF PHYSICS*;
Web-Preprint,

http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[3]
G.W. Bruhn, *Remarks on the "Evans Wave Equation"* ;

http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansWaveEqu.html

To be supplemented occasionally.